Author | Nowinski, J. L. author |
---|---|
Title | Applications of Functional Analysis in Engineering [electronic resource] / by J. L. Nowinski |
Imprint | Boston, MA : Springer US : Imprint: Springer, 1981 |
Connect to | http://dx.doi.org/10.1007/978-1-4684-3926-7 |
Descript | XV, 304 p. online resource |
1. Physical Space. Abstract Spaces -- Comment 1.1 -- 2. Basic Vector Algebra -- Axioms 2.1โ2.3 and Definitions 2.1โ2.3 -- Axioms 2.4โ2.8 -- Theorem 2.1 (Parallelogram Law) -- Problems -- 3. Inner Product of Vectors. Norm -- Definitions 3.1 and 3.2 -- Pythagorean Theorem -- Minkowski Inequality -- CauchyโSchwarz Inequality -- Problems -- 4. Linear Independence. Vector Components. Space Dimension -- Span. Basis. Space Dimension -- Vector Components -- Problems -- 5. Euclidean Spaces of Many Dimensions -- Definitions 5.1โ5.6 -- Definitions 5.7โ5.9 -- Orthogonal Projections -- CauchyโSchwarz and Minkowski Inequalities -- GramโSchmidt Orthogonalization Process -- lpโSpace -- Problems -- 6. Infinite-Dimensional Euclidean Spaces -- Section 6.1. Convergence of a Sequence of Vectors in ?? -- Cauchy Sequence -- Section 6.2. Linear Independence. Span, Basis -- Section 6.3. Linear Manifold -- Subspace -- Distance -- CauchyโSchwarz Inequality -- Remark 6.1 -- Problems -- 7. Abstract Spaces. Hilbert Space -- Linear Vector Space. Axioms -- Inner Product -- Pre-Hilbert Space. Dimension. Completeness. Separability -- Metric Space -- Space Ca?t?b and l1 -- Normed Spaces. Banach Spaces -- Fourier Coefficients -- Besselโs Inequality. Parsevalโs Equality -- Section 7.1. Contraction Mapping -- Problems -- 8. Function Space -- Hilbert, Dirichlet, and Minkowski Products -- Positive Semi-Definite Metric -- Semi-Norm -- Clapeyron Theorem -- RayleighโBetti Theorem -- Linear Differential Operators. Functionals -- Variational Principles -- Bending of Isotropic Plates -- Torsion of Isotropic Bars -- Section 8.1. Theory of Quantum Mechanics -- Problems -- 9. Some Geometry of Function Space -- Translated Subspaces -- Intrinsic and Extrinsic Vectors -- Hyperplanes -- Convexity -- Perpendicularity. Distance -- Orthogonal Projections -- Orthogonal Complement. Direct Sum -- n-Spheres and Hyperspheres -- Balls -- Problems -- 10. Closeness of Functions. Approximation in the Mean. Fourier Expansions -- Uniform Convergence. Mean Square -- Energy Norm -- Space ?2 -- Generalized Fourier Series -- Eigenvalue Problems -- Problems -- 11. Bounds and Inequalities -- Lower and Upper Bounds -- Neumann Problem. Dirichlet Integral -- Dirichlet Problem -- Hypercircle -- Geometrical Illustrations -- Bounds and Approximation in the Mean -- Example 11.1. Torsion of an Anisotropic Bar (Numerical Example) -- Example 11.2. Bounds for Deflection of Anisotropic Plates (Numerical Example) -- Section 11.1. Bounds for a Solution at a Point -- Section 11.1.1. The L*L Method of KatoโFujita -- Poissonโs Problem -- Section 11.1.2. The Diaz-Greenberg Method -- Example 11.3. Bending a Circular Plate (Numerical Example) -- Section 11.1.3. The Washizu Procedure -- Example 11.4. Circular Plate (Numerical Example) -- Problems -- 12. The Method of the Hypercircle -- Elastic State Vector -- Inner Product -- Orthogonal Subspaces -- Uniqueness Theorem -- Vertices -- Hypersphere. Hyperplane. Hypercircle -- Section 12.1. Bounds on an Elastic State -- Fundamental and Auxiliary States -- Example 12.1. Elastic Cylinder in Gravity Field (Numerical Example) -- Galerkin Method -- Section 12.2. Bounds for a Solution at a Point -- Greenโs Function -- Section 12.3. Hypercircle Method and Function Space Inequalities -- Section 12.4. A Comment -- Problems -- 13. The Method of Orthogonal Projections -- Illustrations. Projection Theorem -- Example 13.1. Arithmetic Progression (Numerical Example) -- Example 13.2. A Heated Bar (Numerical Example) -- Section 13.1. Theory of Approximations. Chebyshev Norm -- Example 13.3. Linear Approximation (Numerical Example) -- Problems -- 14. The RayleighโRitz and Trefftz Methods -- Section 14.1. The RayleighโRitz Method -- Coordinate Functions. Admissibility -- Sequences of Functionals -- Lagrange and Castigliano Principles -- Example 14.1. Bounds for Torsional Rigidity -- Example 14.2. Biharmonic Problem -- Section 14.2. The Trefftz Method -- Dirichlet Problem. More General Problem -- Section 14.3. Remark -- Section 14.4. Improvement of Bounds -- Problems -- 15. Function Space and Variational Methods -- Section 15.1. The Inverse Method -- Symmetry and Nondegeneracy of Forms -- Section 15.2. Orthogonal Subspaces -- Minimum Principles -- Section 15.3. Lawsโ Approach -- Reciprocal and Clapeyron Theorems -- Minimum Energy Theorem -- Section 15.4. A Plane Tripod -- Lines of Self-Equilibrated Stress and Equilibrium States -- Minimum Principle -- Maximum Principle -- Problems -- 16. Distributions. Sobolev Spaces -- Section 16.1. Distributions -- Delta Function -- Test Functions -- Functionals -- Distribution -- Differentiation of Distributions -- An Example -- Section 16.2. Sobolev Spaces -- Answers to Problems -- References